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In geometry, a curve of pursuit is a
The pursuit curve was first studied by
The path followed by a single pursuer, following a pursuee that moves at constant speed on a
Typical drawings of curves of pursuit have each point acting as both pursuer and pursuee, inside a polygon, and having each pursuer pursue the adjacent point on the polygon. An example of this is the mice problem.
Mathworld
with a slightly narrower definition that , ''L''′(''t''), and , ''F''′(''t''), are constant
{{Differential transforms of plane curves Curves Pursuit–evasion
In geometry, a curve of pursuit is a curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
constructed by analogy to having a point or points representing pursuers and pursuees; the curve of pursuit is the curve traced by the pursuers.
With the paths of the pursuer and pursuee parameterized in time, the pursuee is always on the pursuer's tangent. That is, given , the pursuer (follower), and , the pursued (leader), for every with there is an such that
:
History
The pursuit curve was first studied by Pierre Bouguer
Pierre Bouguer () (16 February 1698, Croisic – 15 August 1758, Paris) was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture".
Career
Bouguer's father, Jean Bouguer, one ...
in 1732. In an article on navigation, Bouguer defined a curve of pursuit to explore the way in which one ship might maneuver while pursuing another.
Leonardo da Vinci has occasionally been credited with first exploring curves of pursuit. However Paul J. Nahin, having traced such accounts as far back as the late 19th century, indicates that these anecdotes are unfounded.
Single pursuer
The path followed by a single pursuer, following a pursuee that moves at constant speed on a line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
, is a radiodrome In geometry, a radiodrome is the pursuit curve followed by a point that is pursuing another linearly-moving point. The term is derived from the Greek words and . The classic (and best-known) form of a radiodrome is known as the "dog curve"; this i ...
.
It is a solution of the differential equation
.
Multiple pursuers
Typical drawings of curves of pursuit have each point acting as both pursuer and pursuee, inside a polygon, and having each pursuer pursue the adjacent point on the polygon. An example of this is the mice problem.
See also
* Logarithmic spiral * Tractrix * Circles of Apollonius#Apollonius pursuit problem * Pursuit–evasionReferences
External links
Mathworld
with a slightly narrower definition that , ''L''′(''t''), and , ''F''′(''t''), are constant
{{Differential transforms of plane curves Curves Pursuit–evasion